Let S φ T be a mapping of sets. Prove that φis a monomorphism if and only if φ^-1[φ[A]]=A for ev

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Let $S \xrightarrow{\varphi} T$ be a mapping of sets. Prove that $\varphi$ is a monomorphism if and only if $\varphi^{-1}[\varphi[A]]=A$ for every subset $A$ of $S$. Prove that $\varphi$ is an epimorphism if and only if $\varphi\left[\varphi^{-1}[B]\right]=B$ for every subset $B$ of $T$.

   Let $S \xrightarrow{\varphi} T$ be a mapping of sets. Prove that $\varphi$ is a monomorphism if and only if $\varphi^{-1}[\varphi[A]]=A$ for every subset $A$ of $S$. Prove that $\varphi$ is an epimorphism if and only if $\varphi\left[\varphi^{-1}[B]\right]=B$ for every subset $B$ of $T$.

Mathematical physics

Robert Geroch 1st Edition

Chapter 25, Problem 151

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Let $S \xrightarrow{\varphi} T$ be a mapping of sets. Prove that $\varphi$ is a monomorphism if and only if $\varphi^{-1}[\varphi[A]]=A$ for every subset $A$ of $S$. Prove that $\varphi$ is an epimorphism if and only if $\varphi\left[\varphi^{-1}[B]\right]=B$ for every subset $B$ of $T$.

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